LMFDB - Embedded newform 448.4.j.b.335.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(111,448)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("448.111");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.j (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$44$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			335.14
Character	$\chi$	$=$	448.335
Dual form			448.4.j.b.111.14

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.83962 + 2.83962i) q^{3} +(-7.83669 + 7.83669i) q^{5} +(12.7115 + 13.4692i) q^{7} +10.8731i q^{9} +O(q^{10})$	\(q+(-2.83962 + 2.83962i) q^{3} +(-7.83669 + 7.83669i) q^{5} +(12.7115 + 13.4692i) q^{7} +10.8731i q^{9} +(13.0368 - 13.0368i) q^{11} +(40.8123 + 40.8123i) q^{13} -44.5065i q^{15} +83.8552i q^{17} +(12.2047 - 12.2047i) q^{19} +(-74.3432 - 2.15167i) q^{21} +195.456 q^{23} +2.17256i q^{25} +(-107.545 - 107.545i) q^{27} +(-100.285 + 100.285i) q^{29} +66.9646 q^{31} +74.0394i q^{33} +(-205.170 - 5.93810i) q^{35} +(-195.938 - 195.938i) q^{37} -231.783 q^{39} +224.852 q^{41} +(-106.283 + 106.283i) q^{43} +(-85.2089 - 85.2089i) q^{45} -474.554 q^{47} +(-19.8379 + 342.426i) q^{49} +(-238.117 - 238.117i) q^{51} +(57.0387 + 57.0387i) q^{53} +204.331i q^{55} +69.3137i q^{57} +(483.525 + 483.525i) q^{59} +(-447.111 - 447.111i) q^{61} +(-146.451 + 138.213i) q^{63} -639.667 q^{65} +(-475.111 - 475.111i) q^{67} +(-555.022 + 555.022i) q^{69} +197.022 q^{71} -509.792 q^{73} +(-6.16925 - 6.16925i) q^{75} +(341.313 + 9.87841i) q^{77} -196.469i q^{79} +317.204 q^{81} +(-562.513 + 562.513i) q^{83} +(-657.147 - 657.147i) q^{85} -569.546i q^{87} +1068.91 q^{89} +(-30.9247 + 1068.49i) q^{91} +(-190.154 + 190.154i) q^{93} +191.289i q^{95} -295.588i q^{97} +(141.750 + 141.750i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 4 q^{7}+O(q^{10}) $ 88 * q + 4 * q^7	$ 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77} - 1952 q^{81} + 496 q^{85} + 1964 q^{91} - 112 q^{93} - 7592 q^{99}+O(q^{100}) $ 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.83962	+	2.83962i	−0.546486	+	0.546486i	−0.925423	−	0.378937i	$-0.876290\pi$
$3$	−2.83962	+	2.83962i	−0.546486	+	0.546486i	0.378937	+	0.925423i	$0.376290\pi$
$4$		0			0
$5$	−7.83669	+	7.83669i	−0.700935	+	0.700935i	−0.964611	−	0.263676i	$-0.915065\pi$
$5$	−7.83669	+	7.83669i	−0.700935	+	0.700935i	0.263676	+	0.964611i	$0.415065\pi$
$6$		0			0
$7$	12.7115	+	13.4692i	0.686354	+	0.727268i
$7$	12.7115	+	13.4692i	0.686354	+	0.727268i
$8$		0			0
$9$			10.8731i			0.402706i
$10$		0			0
$11$	13.0368	−	13.0368i	0.357341	−	0.357341i	−0.505491	−	0.862832i	$-0.668689\pi$
$11$	13.0368	−	13.0368i	0.357341	−	0.357341i	0.862832	+	0.505491i	$0.168689\pi$
$12$		0			0
$13$	40.8123	+	40.8123i	0.870715	+	0.870715i	0.992550	−	0.121835i	$-0.0388780\pi$
$13$	40.8123	+	40.8123i	0.870715	+	0.870715i	−0.121835	+	0.992550i	$0.538878\pi$
$14$		0			0
$15$		−	44.5065i		−	0.766102i
$16$		0			0
$17$			83.8552i			1.19635i	0.801367	+	0.598173i	$0.204106\pi$
$17$			83.8552i			1.19635i	−0.801367	+	0.598173i	$0.795894\pi$
$18$		0			0
$19$	12.2047	−	12.2047i	0.147366	−	0.147366i	−0.629574	−	0.776940i	$-0.716771\pi$
$19$	12.2047	−	12.2047i	0.147366	−	0.147366i	0.776940	+	0.629574i	$0.216771\pi$
$20$		0			0
$21$	−74.3432	−	2.15167i	−0.772524	−	0.0223587i
$22$		0			0
$23$	195.456			1.77198			0.885988	−	0.463708i	$-0.153481\pi$
$23$	195.456			1.77198			0.885988	+	0.463708i	$0.153481\pi$
$24$		0			0
$25$			2.17256i			0.0173805i
$26$		0			0
$27$	−107.545	−	107.545i	−0.766559	−	0.766559i
$28$		0			0
$29$	−100.285	+	100.285i	−0.642156	+	0.642156i	−0.951085	−	0.308929i	$-0.900029\pi$
$29$	−100.285	+	100.285i	−0.642156	+	0.642156i	0.308929	+	0.951085i	$0.400029\pi$
$30$		0			0
$31$	66.9646			0.387974			0.193987	−	0.981004i	$-0.437858\pi$
$31$	66.9646			0.387974			0.193987	+	0.981004i	$0.437858\pi$
$32$		0			0
$33$			74.0394i			0.390564i
$34$		0			0
$35$	−205.170	−	5.93810i	−0.990857	−	0.0286778i
$36$		0			0
$37$	−195.938	−	195.938i	−0.870595	−	0.870595i	0.121943	−	0.992537i	$-0.461088\pi$
$37$	−195.938	−	195.938i	−0.870595	−	0.870595i	−0.992537	+	0.121943i	$0.961088\pi$
$38$		0			0
$39$	−231.783			−0.951667
$40$		0			0
$41$	224.852			0.856486			0.428243	−	0.903664i	$-0.359133\pi$
$41$	224.852			0.856486			0.428243	+	0.903664i	$0.359133\pi$
$42$		0			0
$43$	−106.283	+	106.283i	−0.376931	+	0.376931i	−0.869994	−	0.493063i	$-0.835877\pi$
$43$	−106.283	+	106.283i	−0.376931	+	0.376931i	0.493063	+	0.869994i	$0.335877\pi$
$44$		0			0
$45$	−85.2089	−	85.2089i	−0.282271	−	0.282271i
$46$		0			0
$47$	−474.554			−1.47278			−0.736392	−	0.676555i	$-0.763472\pi$
$47$	−474.554			−1.47278			−0.736392	+	0.676555i	$0.763472\pi$
$48$		0			0
$49$	−19.8379	+	342.426i	−0.0578364	+	0.998326i
$50$		0			0
$51$	−238.117	−	238.117i	−0.653786	−	0.653786i
$52$		0			0
$53$	57.0387	+	57.0387i	0.147828	+	0.147828i	0.777147	−	0.629319i	$-0.216666\pi$
$53$	57.0387	+	57.0387i	0.147828	+	0.147828i	−0.629319	+	0.777147i	$0.716666\pi$
$54$		0			0
$55$			204.331i			0.500946i
$56$		0			0
$57$			69.3137i			0.161067i
$58$		0			0
$59$	483.525	+	483.525i	1.06694	+	1.06694i	0.997592	+	0.0693491i	$0.0220922\pi$
$59$	483.525	+	483.525i	1.06694	+	1.06694i	0.0693491	+	0.997592i	$0.477908\pi$
$60$		0			0
$61$	−447.111	−	447.111i	−0.938471	−	0.938471i	0.0597425	−	0.998214i	$-0.480972\pi$
$61$	−447.111	−	447.111i	−0.938471	−	0.938471i	−0.998214	+	0.0597425i	$0.980972\pi$
$62$		0			0
$63$	−146.451	+	138.213i	−0.292875	+	0.276399i
$64$		0			0
$65$	−639.667			−1.22063
$66$		0			0
$67$	−475.111	−	475.111i	−0.866328	−	0.866328i	0.125735	−	0.992064i	$-0.459871\pi$
$67$	−475.111	−	475.111i	−0.866328	−	0.866328i	−0.992064	+	0.125735i	$0.959871\pi$
$68$		0			0
$69$	−555.022	+	555.022i	−0.968360	+	0.968360i
$70$		0			0
$71$	197.022			0.329326			0.164663	−	0.986350i	$-0.447346\pi$
$71$	197.022			0.329326			0.164663	+	0.986350i	$0.447346\pi$
$72$		0			0
$73$	−509.792			−0.817351			−0.408676	−	0.912680i	$-0.634009\pi$
$73$	−509.792			−0.817351			−0.408676	+	0.912680i	$0.634009\pi$
$74$		0			0
$75$	−6.16925	−	6.16925i	−0.00949819	−	0.00949819i
$76$		0			0
$77$	341.313	+	9.87841i	0.505145	+	0.0146201i
$78$		0			0
$79$		−	196.469i		−	0.279804i	−0.990165	−	0.139902i	$-0.955321\pi$
$79$		−	196.469i		−	0.279804i	0.990165	−	0.139902i	$-0.0446788\pi$
$80$		0			0
$81$	317.204			0.435121
$82$		0			0
$83$	−562.513	+	562.513i	−0.743902	+	0.743902i	−0.973326	−	0.229424i	$-0.926316\pi$
$83$	−562.513	+	562.513i	−0.743902	+	0.743902i	0.229424	+	0.973326i	$0.426316\pi$
$84$		0			0
$85$	−657.147	−	657.147i	−0.838561	−	0.838561i
$86$		0			0
$87$		−	569.546i		−	0.701859i
$88$		0			0
$89$	1068.91			1.27308			0.636538	−	0.771245i	$-0.280366\pi$
$89$	1068.91			1.27308			0.636538	+	0.771245i	$0.280366\pi$
$90$		0			0
$91$	−30.9247	+	1068.49i	−0.0356241	+	1.23086i
$92$		0			0
$93$	−190.154	+	190.154i	−0.212022	+	0.212022i
$94$		0			0
$95$			191.289i			0.206588i
$96$		0			0
$97$		−	295.588i		−	0.309406i	−0.987961	−	0.154703i	$-0.950558\pi$
$97$		−	295.588i		−	0.309406i	0.987961	−	0.154703i	$-0.0494421\pi$
$98$		0			0
$99$	141.750	+	141.750i	0.143904	+	0.143904i
$100$		0			0
$101$	−642.542	+	642.542i	−0.633023	+	0.633023i	−0.948825	−	0.315802i	$-0.897727\pi$
$101$	−642.542	+	642.542i	−0.633023	+	0.633023i	0.315802	+	0.948825i	$0.397727\pi$
$102$		0			0
$103$		−	1278.13i		−	1.22270i	−0.791361	−	0.611349i	$-0.790627\pi$
$103$		−	1278.13i		−	1.22270i	0.791361	−	0.611349i	$-0.209373\pi$
$104$		0			0
$105$	599.466	−	565.742i	0.557161	−	0.525817i
$106$		0			0
$107$	148.952	−	148.952i	0.134577	−	0.134577i	−0.636610	−	0.771186i	$-0.719664\pi$
$107$	148.952	−	148.952i	0.134577	−	0.134577i	0.771186	+	0.636610i	$0.219664\pi$
$108$		0			0
$109$	1226.33	−	1226.33i	1.07763	−	1.07763i	0.0809036	−	0.996722i	$-0.474219\pi$
$109$	1226.33	−	1226.33i	1.07763	−	1.07763i	0.996722	−	0.0809036i	$-0.0257806\pi$
$110$		0			0
$111$	1112.78			0.951535
$112$		0			0
$113$	−438.708			−0.365223			−0.182611	−	0.983185i	$-0.558455\pi$
$113$	−438.708			−0.365223			−0.182611	+	0.983185i	$0.558455\pi$
$114$		0			0
$115$	−1531.73	+	1531.73i	−1.24204	+	1.24204i
$116$		0			0
$117$	−443.755	+	443.755i	−0.350642	+	0.350642i
$118$		0			0
$119$	−1129.46	+	1065.92i	−0.870064	+	0.821117i
$120$		0			0
$121$			991.082i			0.744615i
$122$		0			0
$123$	−638.494	+	638.494i	−0.468058	+	0.468058i
$124$		0			0
$125$	−996.612	−	996.612i	−0.713118	−	0.713118i
$126$		0			0
$127$		−	353.744i		−	0.247163i	−0.992334	−	0.123582i	$-0.960562\pi$
$127$		−	353.744i		−	0.247163i	0.992334	−	0.123582i	$-0.0394381\pi$
$128$		0			0
$129$		−	603.608i		−	0.411975i
$130$		0			0
$131$	84.6541	−	84.6541i	0.0564600	−	0.0564600i	−0.678313	−	0.734773i	$-0.737289\pi$
$131$	84.6541	−	84.6541i	0.0564600	−	0.0564600i	0.734773	+	0.678313i	$0.237289\pi$
$132$		0			0
$133$	319.527	+	9.24789i	0.208320	+	0.00602928i
$134$		0			0
$135$	1685.60			1.07462
$136$		0			0
$137$			2932.67i			1.82887i	0.404732	+	0.914435i	$0.367365\pi$
$137$			2932.67i			1.82887i	−0.404732	+	0.914435i	$0.632635\pi$
$138$		0			0
$139$	175.424	+	175.424i	0.107045	+	0.107045i	0.758601	−	0.651556i	$-0.225883\pi$
$139$	175.424	+	175.424i	0.107045	+	0.107045i	−0.651556	+	0.758601i	$0.725883\pi$
$140$		0			0
$141$	1347.56	−	1347.56i	0.804856	−	0.804856i
$142$		0			0
$143$	1064.13			0.622285
$144$		0			0
$145$		−	1571.81i		−	0.900219i
$146$		0			0
$147$	−916.029	−	1028.69i	−0.513964	−	0.577178i
$148$		0			0
$149$	−667.066	−	667.066i	−0.366766	−	0.366766i	0.499530	−	0.866296i	$-0.333506\pi$
$149$	−667.066	−	667.066i	−0.366766	−	0.366766i	−0.866296	+	0.499530i	$0.833506\pi$
$150$		0			0
$151$	2452.28			1.32161			0.660806	−	0.750557i	$-0.270215\pi$
$151$	2452.28			1.32161			0.660806	+	0.750557i	$0.270215\pi$
$152$		0			0
$153$	−911.764			−0.481776
$154$		0			0
$155$	−524.781	+	524.781i	−0.271945	+	0.271945i
$156$		0			0
$157$	1779.43	+	1779.43i	0.904549	+	0.904549i	0.995826	−	0.0912765i	$-0.0290947\pi$
$157$	1779.43	+	1779.43i	0.904549	+	0.904549i	−0.0912765	+	0.995826i	$0.529095\pi$
$158$		0			0
$159$	−323.937			−0.161571
$160$		0			0
$161$	2484.53	+	2632.64i	1.21620	+	1.28870i
$162$		0			0
$163$	−522.852	−	522.852i	−0.251245	−	0.251245i	0.570236	−	0.821481i	$-0.306852\pi$
$163$	−522.852	−	522.852i	−0.251245	−	0.251245i	−0.821481	+	0.570236i	$0.806852\pi$
$164$		0			0
$165$	−580.224	−	580.224i	−0.273760	−	0.273760i
$166$		0			0
$167$		−	3423.89i		−	1.58652i	−0.608884	−	0.793259i	$-0.708383\pi$
$167$		−	3423.89i		−	1.58652i	0.608884	−	0.793259i	$-0.291617\pi$
$168$		0			0
$169$			1134.29i			0.516289i
$170$		0			0
$171$	132.703	+	132.703i	0.0593452	+	0.0593452i
$172$		0			0
$173$	−1552.53	−	1552.53i	−0.682294	−	0.682294i	0.278223	−	0.960517i	$-0.410255\pi$
$173$	−1552.53	−	1552.53i	−0.682294	−	0.682294i	−0.960517	+	0.278223i	$0.910255\pi$
$174$		0			0
$175$	−29.2626	+	27.6164i	−0.0126403	+	0.0119292i
$176$		0			0
$177$	−2746.06			−1.16614
$178$		0			0
$179$	2198.60	+	2198.60i	0.918053	+	0.918053i	0.996888	−	0.0788351i	$-0.0251201\pi$
$179$	2198.60	+	2198.60i	0.918053	+	0.918053i	−0.0788351	+	0.996888i	$0.525120\pi$
$180$		0			0
$181$	3032.52	−	3032.52i	1.24533	−	1.24533i	0.287575	−	0.957758i	$-0.407151\pi$
$181$	3032.52	−	3032.52i	1.24533	−	1.24533i	0.957758	−	0.287575i	$-0.0928490\pi$
$182$		0			0
$183$	2539.26			1.02572
$184$		0			0
$185$	3071.01			1.22046
$186$		0			0
$187$	1093.21	+	1093.21i	0.427504	+	0.427504i
$188$		0			0
$189$	81.4903	−	2815.60i	0.0313627	−	1.08362i
$190$		0			0
$191$		−	2242.75i		−	0.849631i	−0.905280	−	0.424815i	$-0.860339\pi$
$191$		−	2242.75i		−	0.849631i	0.905280	−	0.424815i	$-0.139661\pi$
$192$		0			0
$193$	2136.12			0.796690			0.398345	−	0.917236i	$-0.369585\pi$
$193$	2136.12			0.796690			0.398345	+	0.917236i	$0.369585\pi$
$194$		0			0
$195$	1816.41	−	1816.41i	0.667057	−	0.667057i
$196$		0			0
$197$	2179.19	+	2179.19i	0.788125	+	0.788125i	0.981187	−	0.193061i	$-0.0618417\pi$
$197$	2179.19	+	2179.19i	0.788125	+	0.788125i	−0.193061	+	0.981187i	$0.561842\pi$
$198$		0			0
$199$		−	3100.78i		−	1.10457i	−0.833656	−	0.552283i	$-0.813757\pi$
$199$		−	3100.78i		−	1.10457i	0.833656	−	0.552283i	$-0.186243\pi$
$200$		0			0
$201$	2698.27			0.946873
$202$		0			0
$203$	−2625.54	−	75.9893i	−0.907766	−	0.0262729i
$204$		0			0
$205$	−1762.09	+	1762.09i	−0.600341	+	0.600341i
$206$		0			0
$207$			2125.21i			0.713586i
$208$		0			0
$209$		−	318.222i		−	0.105320i
$210$		0			0
$211$	469.715	+	469.715i	0.153254	+	0.153254i	0.779569	−	0.626316i	$-0.215438\pi$
$211$	469.715	+	469.715i	0.153254	+	0.153254i	−0.626316	+	0.779569i	$0.715438\pi$
$212$		0			0
$213$	−559.467	+	559.467i	−0.179972	+	0.179972i
$214$		0			0
$215$		−	1665.82i		−	0.528408i
$216$		0			0
$217$	851.217	+	901.959i	0.266288	+	0.282161i
$218$		0			0
$219$	1447.62	−	1447.62i	0.446671	−	0.446671i
$220$		0			0
$221$	−3422.32	+	3422.32i	−1.04168	+	1.04168i
$222$		0			0
$223$	−2878.06			−0.864256			−0.432128	−	0.901812i	$-0.642237\pi$
$223$	−2878.06			−0.864256			−0.432128	+	0.901812i	$0.642237\pi$
$224$		0			0
$225$	−23.6224			−0.00699923
$226$		0			0
$227$	−1055.07	+	1055.07i	−0.308492	+	0.308492i	−0.844324	−	0.535832i	$-0.819998\pi$
$227$	−1055.07	+	1055.07i	−0.308492	+	0.308492i	0.535832	+	0.844324i	$0.319998\pi$
$228$		0			0
$229$	−2664.88	+	2664.88i	−0.768998	+	0.768998i	−0.977930	−	0.208932i	$-0.933001\pi$
$229$	−2664.88	+	2664.88i	−0.768998	+	0.768998i	0.208932	+	0.977930i	$0.433001\pi$
$230$		0			0
$231$	−997.251	+	941.149i	−0.284044	+	0.268065i
$232$		0			0
$233$			2342.57i			0.658657i	0.944215	+	0.329329i	$0.106822\pi$
$233$			2342.57i			0.658657i	−0.944215	+	0.329329i	$0.893178\pi$
$234$		0			0
$235$	3718.94	−	3718.94i	1.03233	−	1.03233i
$236$		0			0
$237$	557.899	+	557.899i	0.152909	+	0.152909i
$238$		0			0
$239$			3935.33i			1.06509i	0.846403	+	0.532543i	$0.178764\pi$
$239$			3935.33i			1.06509i	−0.846403	+	0.532543i	$0.821236\pi$
$240$		0			0
$241$		−	3237.77i		−	0.865406i	−0.901536	−	0.432703i	$-0.857560\pi$
$241$		−	3237.77i		−	0.865406i	0.901536	−	0.432703i	$-0.142440\pi$
$242$		0			0
$243$	2002.98	−	2002.98i	0.528771	−	0.528771i
$244$		0			0
$245$	−2528.02	−	2838.95i	−0.659222	−	0.740301i
$246$		0			0
$247$	996.206			0.256628
$248$		0			0
$249$		−	3194.65i		−	0.813064i
$250$		0			0
$251$	−3331.16	−	3331.16i	−0.837694	−	0.837694i	0.150861	−	0.988555i	$-0.451795\pi$
$251$	−3331.16	−	3331.16i	−0.837694	−	0.837694i	−0.988555	+	0.150861i	$0.951795\pi$
$252$		0			0
$253$	2548.13	−	2548.13i	0.633200	−	0.633200i
$254$		0			0
$255$	3732.10			0.916523
$256$		0			0
$257$			2061.83i			0.500442i	0.968189	+	0.250221i	$0.0805032\pi$
$257$			2061.83i			0.500442i	−0.968189	+	0.250221i	$0.919497\pi$
$258$		0			0
$259$	148.468	−	5129.78i	0.0356192	−	1.23069i
$260$		0			0
$261$	−1090.41	−	1090.41i	−0.258600	−	0.258600i
$262$		0			0
$263$	3449.44			0.808750			0.404375	−	0.914593i	$-0.367489\pi$
$263$	3449.44			0.808750			0.404375	+	0.914593i	$0.367489\pi$
$264$		0			0
$265$	−893.989			−0.207235
$266$		0			0
$267$	−3035.29	+	3035.29i	−0.695719	+	0.695719i
$268$		0			0
$269$	740.209	+	740.209i	0.167774	+	0.167774i	0.786000	−	0.618226i	$-0.212148\pi$
$269$	740.209	+	740.209i	0.167774	+	0.167774i	−0.618226	+	0.786000i	$0.712148\pi$
$270$		0			0
$271$	−4506.66			−1.01019			−0.505093	−	0.863065i	$-0.668542\pi$
$271$	−4506.66			−1.01019			−0.505093	+	0.863065i	$0.668542\pi$
$272$		0			0
$273$	−2946.30	−	3121.93i	−0.653180	−	0.692117i
$274$		0			0
$275$	28.3233	+	28.3233i	0.00621076	+	0.00621076i
$276$		0			0
$277$	5634.85	+	5634.85i	1.22226	+	1.22226i	0.966827	+	0.255431i	$0.0822173\pi$
$277$	5634.85	+	5634.85i	1.22226	+	1.22226i	0.255431	+	0.966827i	$0.417783\pi$
$278$		0			0
$279$			728.111i			0.156240i
$280$		0			0
$281$		−	3892.55i		−	0.826369i	−0.910647	−	0.413185i	$-0.864416\pi$
$281$		−	3892.55i		−	0.826369i	0.910647	−	0.413185i	$-0.135584\pi$
$282$		0			0
$283$	−449.463	−	449.463i	−0.0944092	−	0.0944092i	0.658325	−	0.752734i	$-0.271265\pi$
$283$	−449.463	−	449.463i	−0.0944092	−	0.0944092i	−0.752734	+	0.658325i	$0.771265\pi$
$284$		0			0
$285$	−543.190	−	543.190i	−0.112897	−	0.112897i
$286$		0			0
$287$	2858.19	+	3028.57i	0.587853	+	0.622894i
$288$		0			0
$289$	−2118.70			−0.431243
$290$		0			0
$291$	839.359	+	839.359i	0.169086	+	0.169086i
$292$		0			0
$293$	1741.54	−	1741.54i	0.347241	−	0.347241i	−0.511840	−	0.859081i	$-0.671036\pi$
$293$	1741.54	−	1741.54i	0.347241	−	0.347241i	0.859081	+	0.511840i	$0.171036\pi$
$294$		0			0
$295$	−7578.47			−1.49571
$296$		0			0
$297$	−2804.10			−0.547846
$298$		0			0
$299$	7977.02	+	7977.02i	1.54289	+	1.54289i
$300$		0			0
$301$	−2782.56	−	80.5339i	−0.532837	−	0.0154216i
$302$		0			0
$303$		−	3649.16i		−	0.691877i
$304$		0			0
$305$	7007.75			1.31561
$306$		0			0
$307$	−3585.35	+	3585.35i	−0.666537	+	0.666537i	−0.956913	−	0.290375i	$-0.906220\pi$
$307$	−3585.35	+	3585.35i	−0.666537	+	0.666537i	0.290375	+	0.956913i	$0.406220\pi$
$308$		0			0
$309$	3629.41	+	3629.41i	0.668188	+	0.668188i
$310$		0			0
$311$			6570.23i			1.19795i	0.800766	+	0.598977i	$0.204426\pi$
$311$			6570.23i			1.19795i	−0.800766	+	0.598977i	$0.795574\pi$
$312$		0			0
$313$	1038.94			0.187619			0.0938093	−	0.995590i	$-0.470096\pi$
$313$	1038.94			0.187619			0.0938093	+	0.995590i	$0.470096\pi$
$314$		0			0
$315$	64.5654	−	2230.82i	0.0115487	−	0.399024i
$316$		0			0
$317$	−2743.50	+	2743.50i	−0.486089	+	0.486089i	−0.907069	−	0.420981i	$-0.861686\pi$
$317$	−2743.50	+	2743.50i	−0.486089	+	0.486089i	0.420981	+	0.907069i	$0.361686\pi$
$318$		0			0
$319$			2614.81i			0.458938i
$320$		0			0
$321$			845.934i			0.147089i
$322$		0			0
$323$	1023.43	+	1023.43i	0.176301	+	0.176301i
$324$		0			0
$325$	−88.6672	+	88.6672i	−0.0151334	+	0.0151334i
$326$		0			0
$327$			6964.64i			1.17781i
$328$		0			0
$329$	−6032.28	−	6391.86i	−1.01085	−	1.07111i
$330$		0			0
$331$	1590.57	−	1590.57i	0.264126	−	0.264126i	−0.562602	−	0.826728i	$-0.690200\pi$
$331$	1590.57	−	1590.57i	0.264126	−	0.264126i	0.826728	+	0.562602i	$0.190200\pi$
$332$		0			0
$333$	2130.45	−	2130.45i	0.350594	−	0.350594i
$334$		0			0
$335$	7446.59			1.21448
$336$		0			0
$337$	8464.78			1.36827			0.684134	−	0.729356i	$-0.260180\pi$
$337$	8464.78			1.36827			0.684134	+	0.729356i	$0.260180\pi$
$338$		0			0
$339$	1245.77	−	1245.77i	0.199589	−	0.199589i
$340$		0			0
$341$	873.006	−	873.006i	0.138639	−	0.138639i
$342$		0			0
$343$	−4864.37	+	4085.53i	−0.765746	+	0.643143i
$344$		0			0
$345$		−	8699.08i		−	1.35751i
$346$		0			0
$347$	708.795	−	708.795i	0.109655	−	0.109655i	−0.650151	−	0.759805i	$-0.725294\pi$
$347$	708.795	−	708.795i	0.109655	−	0.109655i	0.759805	+	0.650151i	$0.225294\pi$
$348$		0			0
$349$	740.620	+	740.620i	0.113595	+	0.113595i	0.761619	−	0.648025i	$-0.224405\pi$
$349$	740.620	+	740.620i	0.113595	+	0.113595i	−0.648025	+	0.761619i	$0.724405\pi$
$350$		0			0
$351$		−	8778.34i		−	1.33491i
$352$		0			0
$353$			5092.33i			0.767811i	0.923372	+	0.383905i	$0.125421\pi$
$353$			5092.33i			0.767811i	−0.923372	+	0.383905i	$0.874579\pi$
$354$		0			0
$355$	−1544.00	+	1544.00i	−0.230836	+	0.230836i
$356$		0			0
$357$	180.429	−	6234.06i	0.0267487	−	0.924206i
$358$		0			0
$359$	−10275.6			−1.51065			−0.755326	−	0.655350i	$-0.772521\pi$
$359$	−10275.6			−1.51065			−0.755326	+	0.655350i	$0.772521\pi$
$360$		0			0
$361$			6561.09i			0.956566i
$362$		0			0
$363$	−2814.30	−	2814.30i	−0.406921	−	0.406921i
$364$		0			0
$365$	3995.08	−	3995.08i	0.572910	−	0.572910i
$366$		0			0
$367$	−13581.8			−1.93179			−0.965895	−	0.258936i	$-0.916628\pi$
$367$	−13581.8			−1.93179			−0.965895	+	0.258936i	$0.916628\pi$
$368$		0			0
$369$			2444.83i			0.344912i
$370$		0			0
$371$	−43.2199	+	1493.31i	−0.00604816	+	0.208972i
$372$		0			0
$373$	−5572.96	−	5572.96i	−0.773611	−	0.773611i	0.205125	−	0.978736i	$-0.434240\pi$
$373$	−5572.96	−	5572.96i	−0.773611	−	0.773611i	−0.978736	+	0.205125i	$0.934240\pi$
$374$		0			0
$375$	5660.01			0.779417
$376$		0			0
$377$	−8185.76			−1.11827
$378$		0			0
$379$	−7185.62	+	7185.62i	−0.973880	+	0.973880i	−0.999667	−	0.0257873i	$-0.991791\pi$
$379$	−7185.62	+	7185.62i	−0.973880	+	0.973880i	0.0257873	+	0.999667i	$0.491791\pi$
$380$		0			0
$381$	1004.50	+	1004.50i	0.135071	+	0.135071i
$382$		0			0
$383$	−424.741			−0.0566665			−0.0283332	−	0.999599i	$-0.509020\pi$
$383$	−424.741			−0.0566665			−0.0283332	+	0.999599i	$0.509020\pi$
$384$		0			0
$385$	−2752.18	+	2597.35i	−0.364322	+	0.343826i
$386$		0			0
$387$	−1155.62	−	1155.62i	−0.151792	−	0.151792i
$388$		0			0
$389$	−2866.08	−	2866.08i	−0.373563	−	0.373563i	0.495210	−	0.868773i	$-0.335091\pi$
$389$	−2866.08	−	2866.08i	−0.373563	−	0.373563i	−0.868773	+	0.495210i	$0.835091\pi$
$390$		0			0
$391$			16390.0i			2.11990i
$392$		0			0
$393$			480.772i			0.0617092i
$394$		0			0
$395$	1539.67	+	1539.67i	0.196125	+	0.196125i
$396$		0			0
$397$	6680.50	+	6680.50i	0.844545	+	0.844545i	0.989446	−	0.144901i	$-0.0462863\pi$
$397$	6680.50	+	6680.50i	0.844545	+	0.844545i	−0.144901	+	0.989446i	$0.546286\pi$
$398$		0			0
$399$	−933.599	+	881.077i	−0.117139	+	0.110549i
$400$		0			0
$401$	−13690.9			−1.70496			−0.852481	−	0.522758i	$-0.824903\pi$
$401$	−13690.9			−1.70496			−0.852481	+	0.522758i	$0.824903\pi$
$402$		0			0
$403$	2732.98	+	2732.98i	0.337815	+	0.337815i
$404$		0			0
$405$	−2485.83	+	2485.83i	−0.304992	+	0.304992i
$406$		0			0
$407$	−5108.82			−0.622199
$408$		0			0
$409$	−12605.4			−1.52395			−0.761976	−	0.647606i	$-0.775770\pi$
$409$	−12605.4			−1.52395			−0.761976	+	0.647606i	$0.775770\pi$
$410$		0			0
$411$	−8327.69	−	8327.69i	−0.999452	−	0.999452i
$412$		0			0
$413$	−366.381	+	12659.0i	−0.0436524	+	1.50825i
$414$		0			0
$415$		−	8816.49i		−	1.04285i
$416$		0			0
$417$	−996.276			−0.116997
$418$		0			0
$419$	6216.37	−	6216.37i	0.724796	−	0.724796i	−0.244782	−	0.969578i	$-0.578716\pi$
$419$	6216.37	−	6216.37i	0.724796	−	0.724796i	0.969578	+	0.244782i	$0.0787164\pi$
$420$		0			0
$421$	4923.03	+	4923.03i	0.569914	+	0.569914i	0.932104	−	0.362191i	$-0.117971\pi$
$421$	4923.03	+	4923.03i	0.569914	+	0.569914i	−0.362191	+	0.932104i	$0.617971\pi$
$422$		0			0
$423$		−	5159.86i		−	0.593099i
$424$		0			0
$425$	−182.181			−0.0207931
$426$		0			0
$427$	338.790	−	11705.7i	0.0383962	−	1.32664i
$428$		0			0
$429$	−3021.72	+	3021.72i	−0.340070	+	0.340070i
$430$		0			0
$431$			9987.61i			1.11621i	0.829770	+	0.558105i	$0.188471\pi$
$431$			9987.61i			1.11621i	−0.829770	+	0.558105i	$0.811529\pi$
$432$		0			0
$433$			2707.71i			0.300518i	0.988647	+	0.150259i	$0.0480107\pi$
$433$			2707.71i			0.300518i	−0.988647	+	0.150259i	$0.951989\pi$
$434$		0			0
$435$	4463.35	+	4463.35i	0.491957	+	0.491957i
$436$		0			0
$437$	2385.49	−	2385.49i	0.261129	−	0.261129i
$438$		0			0
$439$		−	9495.73i		−	1.03236i	−0.856480	−	0.516181i	$-0.827353\pi$
$439$		−	9495.73i		−	1.03236i	0.856480	−	0.516181i	$-0.172647\pi$
$440$		0			0
$441$	−3723.22	−	215.699i	−0.402032	−	0.0232911i
$442$		0			0
$443$	−1336.57	+	1336.57i	−0.143347	+	0.143347i	−0.775138	−	0.631792i	$-0.782320\pi$
$443$	−1336.57	+	1336.57i	−0.143347	+	0.143347i	0.631792	+	0.775138i	$0.282320\pi$
$444$		0			0
$445$	−8376.69	+	8376.69i	−0.892344	+	0.892344i
$446$		0			0
$447$	3788.43			0.400865
$448$		0			0
$449$	−17752.5			−1.86591			−0.932953	−	0.359999i	$-0.882777\pi$
$449$	−17752.5			−1.86591			−0.932953	+	0.359999i	$0.882777\pi$
$450$		0			0
$451$	2931.35	−	2931.35i	0.306058	−	0.306058i
$452$		0			0
$453$	−6963.55	+	6963.55i	−0.722243	+	0.722243i
$454$		0			0
$455$	−8131.09	−	8615.79i	−0.837784	−	0.887724i
$456$		0			0
$457$		−	5301.67i		−	0.542673i	−0.962485	−	0.271336i	$-0.912534\pi$
$457$		−	5301.67i		−	0.542673i	0.962485	−	0.271336i	$-0.0874656\pi$
$458$		0			0
$459$	9018.23	−	9018.23i	0.917070	−	0.917070i
$460$		0			0
$461$	5484.66	+	5484.66i	0.554113	+	0.554113i	0.927625	−	0.373512i	$-0.121847\pi$
$461$	5484.66	+	5484.66i	0.554113	+	0.554113i	−0.373512	+	0.927625i	$0.621847\pi$
$462$		0			0
$463$		−	13015.7i		−	1.30646i	−0.757161	−	0.653228i	$-0.773414\pi$
$463$		−	13015.7i		−	1.30646i	0.757161	−	0.653228i	$-0.226586\pi$
$464$		0			0
$465$		−	2980.36i		−	0.297228i
$466$		0			0
$467$	−2115.14	+	2115.14i	−0.209587	+	0.209587i	−0.804092	−	0.594505i	$-0.797348\pi$
$467$	−2115.14	+	2115.14i	−0.209587	+	0.209587i	0.594505	+	0.804092i	$0.297348\pi$
$468$		0			0
$469$	360.006	−	12438.7i	0.0354446	−	1.22466i
$470$		0			0
$471$	−10105.8			−0.988647
$472$		0			0
$473$			2771.19i			0.269386i
$474$		0			0
$475$	26.5155	+	26.5155i	0.00256129	+	0.00256129i
$476$		0			0
$477$	−620.185	+	620.185i	−0.0595311	+	0.0595311i
$478$		0			0
$479$	10959.2			1.04539			0.522693	−	0.852521i	$-0.324928\pi$
$479$	10959.2			1.04539			0.522693	+	0.852521i	$0.324928\pi$
$480$		0			0
$481$		−	15993.4i		−	1.51608i
$482$		0			0
$483$	−14530.8	−	420.557i	−1.36889	−	0.0396191i
$484$		0			0
$485$	2316.43	+	2316.43i	0.216874	+	0.216874i
$486$		0			0
$487$	477.058			0.0443893			0.0221946	−	0.999754i	$-0.492935\pi$
$487$	477.058			0.0443893			0.0221946	+	0.999754i	$0.492935\pi$
$488$		0			0
$489$	2969.41			0.274604
$490$		0			0
$491$	3116.41	−	3116.41i	0.286439	−	0.286439i	−0.549231	−	0.835670i	$-0.685079\pi$
$491$	3116.41	−	3116.41i	0.286439	−	0.286439i	0.835670	+	0.549231i	$0.185079\pi$
$492$		0			0
$493$	−8409.46	−	8409.46i	−0.768241	−	0.768241i
$494$		0			0
$495$	−2221.71			−0.201734
$496$		0			0
$497$	2504.43	+	2653.72i	0.226034	+	0.239508i
$498$		0			0
$499$	−2044.54	−	2044.54i	−0.183419	−	0.183419i	0.609425	−	0.792844i	$-0.291400\pi$
$499$	−2044.54	−	2044.54i	−0.183419	−	0.183419i	−0.792844	+	0.609425i	$0.791400\pi$
$500$		0			0
$501$	9722.56	+	9722.56i	0.867010	+	0.867010i
$502$		0			0
$503$			10769.7i			0.954665i	0.878723	+	0.477332i	$0.158396\pi$
$503$			10769.7i			0.954665i	−0.878723	+	0.477332i	$0.841604\pi$
$504$		0			0
$505$		−	10070.8i		−	0.887416i
$506$		0			0
$507$	−3220.95	−	3220.95i	−0.282145	−	0.282145i
$508$		0			0
$509$	7925.35	+	7925.35i	0.690148	+	0.690148i	0.962264	−	0.272117i	$-0.0877236\pi$
$509$	7925.35	+	7925.35i	0.690148	+	0.690148i	−0.272117	+	0.962264i	$0.587724\pi$
$510$		0			0
$511$	−6480.20	−	6866.48i	−0.560992	−	0.594433i
$512$		0			0
$513$	−2625.12			−0.225930
$514$		0			0
$515$	10016.3	+	10016.3i	0.857032	+	0.857032i
$516$		0			0
$517$	−6186.69	+	6186.69i	−0.526286	+	0.526286i
$518$		0			0
$519$	8817.22			0.745728
$520$		0			0
$521$	20268.2			1.70435			0.852173	−	0.523260i	$-0.175284\pi$
$521$	20268.2			1.70435			0.852173	+	0.523260i	$0.175284\pi$
$522$		0			0
$523$	−1119.71	−	1119.71i	−0.0936165	−	0.0936165i	0.658748	−	0.752364i	$-0.271087\pi$
$523$	−1119.71	−	1119.71i	−0.0936165	−	0.0936165i	−0.752364	+	0.658748i	$0.771087\pi$
$524$		0			0
$525$	4.67463	−	161.515i	0.000388605	−	0.0134268i
$526$		0			0
$527$			5615.33i			0.464151i
$528$		0			0
$529$	26036.2			2.13990
$530$		0			0
$531$	−5257.40	+	5257.40i	−0.429664	+	0.429664i
$532$		0			0
$533$	9176.71	+	9176.71i	0.745755	+	0.745755i
$534$		0			0
$535$			2334.58i			0.188659i
$536$		0			0
$537$	−12486.4			−1.00341
$538$		0			0
$539$	4205.53	+	4722.77i	0.336076	+	0.377410i
$540$		0			0
$541$	11922.1	−	11922.1i	0.947453	−	0.947453i	−0.0512336	−	0.998687i	$-0.516315\pi$
$541$	11922.1	−	11922.1i	0.947453	−	0.947453i	0.998687	+	0.0512336i	$0.0163153\pi$
$542$		0			0
$543$			17222.4i			1.36111i
$544$		0			0
$545$			19220.7i			1.51069i
$546$		0			0
$547$	14500.3	+	14500.3i	1.13343	+	1.13343i	0.989602	+	0.143833i	$0.0459427\pi$
$547$	14500.3	+	14500.3i	1.13343	+	1.13343i	0.143833	+	0.989602i	$0.454057\pi$
$548$		0			0
$549$	4861.47	−	4861.47i	0.377928	−	0.377928i
$550$		0			0
$551$			2447.91i			0.189264i
$552$		0			0
$553$	2646.28	−	2497.41i	0.203493	−	0.192045i
$554$		0			0
$555$	−8720.51	+	8720.51i	−0.666964	+	0.666964i
$556$		0			0
$557$	1701.60	−	1701.60i	0.129442	−	0.129442i	−0.639418	−	0.768860i	$-0.720824\pi$
$557$	1701.60	−	1701.60i	0.129442	−	0.129442i	0.768860	+	0.639418i	$0.220824\pi$
$558$		0			0
$559$	−8675.31			−0.656398
$560$		0			0
$561$	−6208.59			−0.467249
$562$		0			0
$563$	15742.2	−	15742.2i	1.17843	−	1.17843i	0.198284	−	0.980145i	$-0.436463\pi$
$563$	15742.2	−	15742.2i	1.17843	−	1.17843i	0.980145	−	0.198284i	$-0.0635368\pi$
$564$		0			0
$565$	3438.02	−	3438.02i	0.255997	−	0.255997i
$566$		0			0
$567$	4032.12	+	4272.47i	0.298647	+	0.316450i
$568$		0			0
$569$			4732.59i			0.348683i	0.984685	+	0.174341i	$0.0557796\pi$
$569$			4732.59i			0.348683i	−0.984685	+	0.174341i	$0.944220\pi$
$570$		0			0
$571$	9206.67	−	9206.67i	0.674758	−	0.674758i	−0.284051	−	0.958809i	$-0.591678\pi$
$571$	9206.67	−	9206.67i	0.674758	−	0.674758i	0.958809	+	0.284051i	$0.0916785\pi$
$572$		0			0
$573$	6368.56	+	6368.56i	0.464311	+	0.464311i
$574$		0			0
$575$			424.641i			0.0307978i
$576$		0			0
$577$		−	2080.35i		−	0.150097i	−0.997180	−	0.0750486i	$-0.976089\pi$
$577$		−	2080.35i		−	0.150097i	0.997180	−	0.0750486i	$-0.0239112\pi$
$578$		0			0
$579$	−6065.78	+	6065.78i	−0.435380	+	0.435380i
$580$		0			0
$581$	−14727.0	−	426.234i	−1.05160	−	0.0304357i
$582$		0			0
$583$	1487.21			0.105650
$584$		0			0
$585$		−	6955.14i		−	0.491555i
$586$		0			0
$587$	15882.4	+	15882.4i	1.11676	+	1.11676i	0.992214	+	0.124545i	$0.0397470\pi$
$587$	15882.4	+	15882.4i	1.11676	+	1.11676i	0.124545	+	0.992214i	$0.460253\pi$
$588$		0			0
$589$	817.285	−	817.285i	0.0571742	−	0.0571742i
$590$		0			0
$591$	−12376.1			−0.861399
$592$		0			0
$593$		−	12159.5i		−	0.842040i	−0.907051	−	0.421020i	$-0.861672\pi$
$593$		−	12159.5i		−	0.842040i	0.907051	−	0.421020i	$-0.138328\pi$
$594$		0			0
$595$	497.941	−	17204.5i	0.0343085	−	1.18541i
$596$		0			0
$597$	8805.06	+	8805.06i	0.603630	+	0.603630i
$598$		0			0
$599$	7073.49			0.482496			0.241248	−	0.970463i	$-0.422443\pi$
$599$	7073.49			0.482496			0.241248	+	0.970463i	$0.422443\pi$
$600$		0			0
$601$	7173.12			0.486852			0.243426	−	0.969920i	$-0.421729\pi$
$601$	7173.12			0.486852			0.243426	+	0.969920i	$0.421729\pi$
$602$		0			0
$603$	5165.91	−	5165.91i	0.348876	−	0.348876i
$604$		0			0
$605$	−7766.80	−	7766.80i	−0.521926	−	0.521926i
$606$		0			0
$607$	16083.6			1.07548			0.537738	−	0.843112i	$-0.319279\pi$
$607$	16083.6			1.07548			0.537738	+	0.843112i	$0.319279\pi$
$608$		0			0
$609$	7671.32	−	7239.75i	0.510439	−	0.481724i
$610$		0			0
$611$	−19367.7	−	19367.7i	−1.28238	−	1.28238i
$612$		0			0
$613$	−6214.31	−	6214.31i	−0.409451	−	0.409451i	0.472096	−	0.881547i	$-0.343498\pi$
$613$	−6214.31	−	6214.31i	−0.409451	−	0.409451i	−0.881547	+	0.472096i	$0.843498\pi$
$614$		0			0
$615$		−	10007.4i		−	0.656156i
$616$		0			0
$617$			17264.3i			1.12647i	0.826296	+	0.563237i	$0.190444\pi$
$617$			17264.3i			1.12647i	−0.826296	+	0.563237i	$0.809556\pi$
$618$		0			0
$619$	446.948	+	446.948i	0.0290216	+	0.0290216i	0.721469	−	0.692447i	$-0.243467\pi$
$619$	446.948	+	446.948i	0.0290216	+	0.0290216i	−0.692447	+	0.721469i	$0.743467\pi$
$620$		0			0
$621$	−21020.4	−	21020.4i	−1.35832	−	1.35832i
$622$		0			0
$623$	13587.4	+	14397.3i	0.873781	+	0.925868i
$624$		0			0
$625$	15348.7			0.982317
$626$		0			0
$627$	903.631	+	903.631i	0.0575559	+	0.0575559i
$628$		0			0
$629$	16430.4	−	16430.4i	1.04153	−	1.04153i
$630$		0			0
$631$	19778.1			1.24779			0.623895	−	0.781508i	$-0.285549\pi$
$631$	19778.1			1.24779			0.623895	+	0.781508i	$0.285549\pi$
$632$		0			0
$633$	−2667.63			−0.167502
$634$		0			0
$635$	2772.18	+	2772.18i	0.173245	+	0.173245i
$636$		0			0
$637$	−14784.8	+	13165.6i	−0.919616	+	0.818899i
$638$		0			0
$639$			2142.23i			0.132622i
$640$		0			0
$641$	27822.6			1.71439			0.857197	−	0.514989i	$-0.172204\pi$
$641$	27822.6			1.71439			0.857197	+	0.514989i	$0.172204\pi$
$642$		0			0
$643$	9245.46	−	9245.46i	0.567038	−	0.567038i	−0.364260	−	0.931297i	$-0.618678\pi$
$643$	9245.46	−	9245.46i	0.567038	−	0.567038i	0.931297	+	0.364260i	$0.118678\pi$
$644$		0			0
$645$	4730.29	+	4730.29i	0.288767	+	0.288767i
$646$		0			0
$647$			3653.57i			0.222004i	0.993820	+	0.111002i	$0.0354060\pi$
$647$			3653.57i			0.222004i	−0.993820	+	0.111002i	$0.964594\pi$
$648$		0			0
$649$	12607.3			0.762524
$650$		0			0
$651$	−4978.36	−	144.086i	−0.299719	−	0.00867460i
$652$		0			0
$653$	3761.12	−	3761.12i	0.225397	−	0.225397i	−0.585370	−	0.810767i	$-0.699051\pi$
$653$	3761.12	−	3761.12i	0.225397	−	0.225397i	0.810767	+	0.585370i	$0.199051\pi$
$654$		0			0
$655$			1326.82i			0.0791496i
$656$		0			0
$657$		−	5543.00i		−	0.329152i
$658$		0			0
$659$	13087.9	+	13087.9i	0.773643	+	0.773643i	0.978741	−	0.205098i	$-0.0657513\pi$
$659$	13087.9	+	13087.9i	0.773643	+	0.773643i	−0.205098	+	0.978741i	$0.565751\pi$
$660$		0			0
$661$	−5429.67	+	5429.67i	−0.319500	+	0.319500i	−0.848575	−	0.529075i	$-0.822539\pi$
$661$	−5429.67	+	5429.67i	−0.319500	+	0.319500i	0.529075	+	0.848575i	$0.322539\pi$
$662$		0			0
$663$		−	19436.2i		−	1.13852i
$664$		0			0
$665$	−2576.51	+	2431.57i	−0.150245	+	0.141793i
$666$		0			0
$667$	−19601.4	+	19601.4i	−1.13789	+	1.13789i
$668$		0			0
$669$	8172.60	−	8172.60i	0.472304	−	0.472304i
$670$		0			0
$671$	−11657.8			−0.670709
$672$		0			0
$673$	−33268.8			−1.90552			−0.952762	−	0.303718i	$-0.901772\pi$
$673$	−33268.8			−1.90552			−0.952762	+	0.303718i	$0.901772\pi$
$674$		0			0
$675$	233.649	−	233.649i	0.0133232	−	0.0133232i
$676$		0			0
$677$	16950.8	−	16950.8i	0.962293	−	0.962293i	−0.0370215	−	0.999314i	$-0.511787\pi$
$677$	16950.8	−	16950.8i	0.962293	−	0.962293i	0.999314	+	0.0370215i	$0.0117870\pi$
$678$		0			0
$679$	3981.33	−	3757.36i	0.225021	−	0.212362i
$680$		0			0
$681$		−	5992.03i		−	0.337173i
$682$		0			0
$683$	−11603.7	+	11603.7i	−0.650080	+	0.650080i	−0.953012	−	0.302932i	$-0.902034\pi$
$683$	−11603.7	+	11603.7i	−0.650080	+	0.650080i	0.302932	+	0.953012i	$0.402034\pi$
$684$		0			0
$685$	−22982.5	−	22982.5i	−1.28192	−	1.28192i
$686$		0			0
$687$		−	15134.5i		−	0.840493i
$688$		0			0
$689$			4655.76i			0.257431i
$690$		0			0
$691$	11331.4	−	11331.4i	0.623830	−	0.623830i	−0.322679	−	0.946509i	$-0.604583\pi$
$691$	11331.4	−	11331.4i	0.623830	−	0.623830i	0.946509	+	0.322679i	$0.104583\pi$
$692$		0			0
$693$	−107.409	+	3711.12i	−0.00588761	+	0.203425i
$694$		0			0
$695$	−2749.49			−0.150063
$696$		0			0
$697$			18855.0i			1.02465i
$698$		0			0
$699$	−6652.03	−	6652.03i	−0.359947	−	0.359947i
$700$		0			0
$701$	4647.57	−	4647.57i	0.250408	−	0.250408i	−0.570730	−	0.821138i	$-0.693340\pi$
$701$	4647.57	−	4647.57i	0.250408	−	0.250408i	0.821138	+	0.570730i	$0.193340\pi$
$702$		0			0
$703$	−4782.74			−0.256592
$704$		0			0
$705$			21120.8i			1.12830i
$706$		0			0
$707$	−16822.2	−	486.874i	−0.894855	−	0.0258993i
$708$		0			0
$709$	−23636.9	−	23636.9i	−1.25205	−	1.25205i	−0.954800	−	0.297249i	$-0.903931\pi$
$709$	−23636.9	−	23636.9i	−1.25205	−	1.25205i	−0.297249	−	0.954800i	$-0.596069\pi$
$710$		0			0
$711$	2136.22			0.112679
$712$		0			0
$713$	13088.7			0.687481
$714$		0			0
$715$	−8339.23	+	8339.23i	−0.436181	+	0.436181i
$716$		0			0
$717$	−11174.9	−	11174.9i	−0.582054	−	0.582054i
$718$		0			0
$719$	10110.1			0.524398			0.262199	−	0.965014i	$-0.415552\pi$
$719$	10110.1			0.524398			0.262199	+	0.965014i	$0.415552\pi$
$720$		0			0
$721$	17215.4	−	16246.9i	0.889229	−	0.839204i
$722$		0			0
$723$	9194.04	+	9194.04i	0.472932	+	0.472932i
$724$		0			0
$725$	−217.876	−	217.876i	−0.0111610	−	0.0111610i
$726$		0			0
$727$		−	29725.4i		−	1.51644i	−0.651997	−	0.758221i	$-0.726069\pi$
$727$		−	29725.4i		−	1.51644i	0.651997	−	0.758221i	$-0.273931\pi$
$728$		0			0
$729$			19939.9i			1.01305i
$730$		0			0
$731$	−8912.39	−	8912.39i	−0.450939	−	0.450939i
$732$		0			0
$733$	16917.5	+	16917.5i	0.852473	+	0.852473i	0.990437	−	0.137964i	$-0.0440559\pi$
$733$	16917.5	+	16917.5i	0.852473	+	0.852473i	−0.137964	+	0.990437i	$0.544056\pi$
$734$		0			0
$735$	15240.2	+	882.914i	0.764820	+	0.0443086i
$736$		0			0
$737$	−12387.9			−0.619150
$738$		0			0
$739$	−11088.2	−	11088.2i	−0.551943	−	0.551943i	0.375058	−	0.927001i	$-0.377623\pi$
$739$	−11088.2	−	11088.2i	−0.551943	−	0.551943i	−0.927001	+	0.375058i	$0.877623\pi$
$740$		0			0
$741$	−2828.85	+	2828.85i	−0.140243	+	0.140243i
$742$		0			0
$743$	31905.6			1.57537			0.787686	−	0.616077i	$-0.211279\pi$
$743$	31905.6			1.57537			0.787686	+	0.616077i	$0.211279\pi$
$744$		0			0
$745$	10455.2			0.514158
$746$		0			0
$747$	−6116.25	−	6116.25i	−0.299574	−	0.299574i
$748$		0			0
$749$	3899.65	+	112.865i	0.190241	+	0.00550602i
$750$		0			0
$751$		−	2629.94i		−	0.127787i	−0.997957	−	0.0638935i	$-0.979648\pi$
$751$		−	2629.94i		−	0.127787i	0.997957	−	0.0638935i	$-0.0203518\pi$
$752$		0			0
$753$	18918.5			0.915576
$754$		0			0
$755$	−19217.7	+	19217.7i	−0.926364	+	0.926364i
$756$		0			0
$757$	5032.50	+	5032.50i	0.241624	+	0.241624i	0.817522	−	0.575898i	$-0.195347\pi$
$757$	5032.50	+	5032.50i	0.241624	+	0.241624i	−0.575898	+	0.817522i	$0.695347\pi$
$758$		0			0
$759$			14471.5i			0.692070i
$760$		0			0
$761$	−8502.82			−0.405029			−0.202515	−	0.979279i	$-0.564911\pi$
$761$	−8502.82			−0.405029			−0.202515	+	0.979279i	$0.564911\pi$
$762$		0			0
$763$	32106.1	+	929.228i	1.52335	+	0.0440895i
$764$		0			0
$765$	7145.21	−	7145.21i	0.337694	−	0.337694i
$766$		0			0
$767$			39467.5i			1.85800i
$768$		0			0
$769$		−	17195.0i		−	0.806329i	−0.915128	−	0.403164i	$-0.867910\pi$
$769$		−	17195.0i		−	0.806329i	0.915128	−	0.403164i	$-0.132090\pi$
$770$		0			0
$771$	−5854.83	−	5854.83i	−0.273484	−	0.273484i
$772$		0			0
$773$	−11063.4	+	11063.4i	−0.514776	+	0.514776i	−0.915986	−	0.401210i	$-0.868590\pi$
$773$	−11063.4	+	11063.4i	−0.514776	+	0.514776i	0.401210	+	0.915986i	$0.368590\pi$
$774$		0			0
$775$			145.485i			0.00674318i
$776$		0			0
$777$	14145.1	+	14988.2i	0.653090	+	0.692021i
$778$		0			0
$779$	2744.25	−	2744.25i	0.126217	−	0.126217i
$780$		0			0
$781$	2568.54	−	2568.54i	0.117682	−	0.117682i
$782$		0			0
$783$	21570.4			0.984502
$784$		0			0
$785$	−27889.7			−1.26806
$786$		0			0
$787$	−22110.7	+	22110.7i	−1.00148	+	1.00148i	−0.00147792	+	0.999999i	$0.500470\pi$
$787$	−22110.7	+	22110.7i	−1.00148	+	1.00148i	−0.999999	+	0.00147792i	$0.999530\pi$
$788$		0			0
$789$	−9795.10	+	9795.10i	−0.441971	+	0.441971i
$790$		0			0
$791$	−5576.62	−	5909.04i	−0.250672	−	0.265615i
$792$		0			0
$793$		−	36495.3i		−	1.63428i
$794$		0			0
$795$	2538.59	−	2538.59i	0.113251	−	0.113251i
$796$		0			0
$797$	6976.76	+	6976.76i	0.310075	+	0.310075i	0.844938	−	0.534864i	$-0.179637\pi$
$797$	6976.76	+	6976.76i	0.310075	+	0.310075i	−0.534864	+	0.844938i	$0.679637\pi$
$798$		0			0
$799$		−	39793.9i		−	1.76196i
$800$		0			0
$801$			11622.3i			0.512676i
$802$		0			0
$803$	−6646.07	+	6646.07i	−0.292073	+	0.292073i
$804$		0			0
$805$	−40101.7	−	1160.64i	−1.75577	−	0.0508163i
$806$		0			0
$807$	−4203.83			−0.183373
$808$		0			0
$809$			31190.3i			1.35549i	0.735296	+	0.677746i	$0.237043\pi$
$809$			31190.3i			1.35549i	−0.735296	+	0.677746i	$0.762957\pi$
$810$		0			0
$811$	1864.47	+	1864.47i	0.0807282	+	0.0807282i	0.746318	−	0.665590i	$-0.231820\pi$
$811$	1864.47	+	1864.47i	0.0807282	+	0.0807282i	−0.665590	+	0.746318i	$0.731820\pi$
$812$		0			0
$813$	12797.2	−	12797.2i	0.552052	−	0.552052i
$814$		0			0
$815$	8194.86			0.352213
$816$		0			0
$817$			2594.31i			0.111094i
$818$		0			0
$819$	−11617.8	−	336.247i	−0.495676	−	0.0143460i
$820$		0			0
$821$	24776.8	+	24776.8i	1.05325	+	1.05325i	0.998500	+	0.0547470i	$0.0174352\pi$
$821$	24776.8	+	24776.8i	1.05325	+	1.05325i	0.0547470	+	0.998500i	$0.482565\pi$
$822$		0			0
$823$	3213.67			0.136114			0.0680568	−	0.997681i	$-0.478320\pi$
$823$	3213.67			0.136114			0.0680568	+	0.997681i	$0.478320\pi$
$824$		0			0
$825$	−160.855			−0.00678819
$826$		0			0
$827$	−21897.0	+	21897.0i	−0.920719	+	0.920719i	−0.997080	−	0.0763610i	$-0.975670\pi$
$827$	−21897.0	+	21897.0i	−0.920719	+	0.920719i	0.0763610	+	0.997080i	$0.475670\pi$
$828$		0			0
$829$	23724.7	+	23724.7i	0.993958	+	0.993958i	0.999982	−	0.00602374i	$-0.00191743\pi$
$829$	23724.7	+	23724.7i	0.993958	+	0.993958i	−0.00602374	+	0.999982i	$0.501917\pi$
$830$		0			0
$831$	−32001.7			−1.33589
$832$		0			0
$833$	−28714.2	−	1663.51i	−1.19434	−	0.0691923i
$834$		0			0
$835$	26832.0	+	26832.0i	1.11205	+	1.11205i
$836$		0			0
$837$	−7201.73	−	7201.73i	−0.297405	−	0.297405i
$838$		0			0
$839$		−	17227.6i		−	0.708896i	−0.935076	−	0.354448i	$-0.884669\pi$
$839$		−	17227.6i		−	0.708896i	0.935076	−	0.354448i	$-0.115331\pi$
$840$		0			0
$841$			4274.68i			0.175271i
$842$		0			0
$843$	11053.4	+	11053.4i	0.451599	+	0.451599i
$844$		0			0
$845$	−8889.06	−	8889.06i	−0.361885	−	0.361885i
$846$		0			0
$847$	−13349.1	+	12598.1i	−0.541534	+	0.511069i
$848$		0			0
$849$	2552.61			0.103187
$850$		0			0
$851$	−38297.3	−	38297.3i	−1.54267	−	1.54267i
$852$		0			0
$853$	−26670.2	+	26670.2i	−1.07054	+	1.07054i	−0.0732233	+	0.997316i	$0.523329\pi$
$853$	−26670.2	+	26670.2i	−1.07054	+	1.07054i	−0.997316	+	0.0732233i	$0.976671\pi$
$854$		0			0
$855$	−2079.90			−0.0831943
$856$		0			0
$857$	22489.6			0.896418			0.448209	−	0.893929i	$-0.352062\pi$
$857$	22489.6			0.896418			0.448209	+	0.893929i	$0.352062\pi$
$858$		0			0
$859$	21711.2	+	21711.2i	0.862371	+	0.862371i	0.991613	−	0.129242i	$-0.0412546\pi$
$859$	21711.2	+	21711.2i	0.862371	+	0.862371i	−0.129242	+	0.991613i	$0.541255\pi$
$860$		0			0
$861$	−16716.2	−	483.806i	−0.661656	−	0.0191499i
$862$		0			0
$863$		−	23367.7i		−	0.921720i	−0.887473	−	0.460860i	$-0.847541\pi$
$863$		−	23367.7i		−	0.921720i	0.887473	−	0.460860i	$-0.152459\pi$
$864$		0			0
$865$	24333.4			0.956487
$866$		0			0
$867$	6016.31	−	6016.31i	0.235668	−	0.235668i
$868$		0			0
$869$	−2561.34	−	2561.34i	−0.0999856	−	0.0999856i
$870$		0			0
$871$		−	38780.7i		−	1.50865i
$872$		0			0
$873$	3213.95			0.124600
$874$		0			0
$875$	755.163	−	26091.9i	0.0291762	−	1.00808i
$876$		0			0
$877$	17547.3	−	17547.3i	0.675635	−	0.675635i	−0.283375	−	0.959009i	$-0.591454\pi$
$877$	17547.3	−	17547.3i	0.675635	−	0.675635i	0.959009	+	0.283375i	$0.0914540\pi$
$878$		0			0
$879$			9890.61i			0.379525i
$880$		0			0
$881$		−	34359.5i		−	1.31396i	−0.753907	−	0.656981i	$-0.771833\pi$
$881$		−	34359.5i		−	1.31396i	0.753907	−	0.656981i	$-0.228167\pi$
$882$		0			0
$883$	8494.08	+	8494.08i	0.323724	+	0.323724i	0.850194	−	0.526470i	$-0.176485\pi$
$883$	8494.08	+	8494.08i	0.323724	+	0.323724i	−0.526470	+	0.850194i	$0.676485\pi$
$884$		0			0
$885$	21520.0	−	21520.0i	0.817386	−	0.817386i
$886$		0			0
$887$			28862.5i			1.09257i	0.837600	+	0.546284i	$0.183958\pi$
$887$			28862.5i			1.09257i	−0.837600	+	0.546284i	$0.816042\pi$
$888$		0			0
$889$	4764.64	−	4496.60i	0.179754	−	0.169641i
$890$		0			0
$891$	4135.33	−	4135.33i	0.155487	−	0.155487i
$892$		0			0
$893$	−5791.80	+	5791.80i	−0.217038	+	0.217038i
$894$		0			0
$895$	−34459.6			−1.28699
$896$		0			0
$897$	−45303.5			−1.68633
$898$		0			0
$899$	−6715.57	+	6715.57i	−0.249140	+	0.249140i
$900$		0			0
$901$	−4782.99	+	4782.99i	−0.176853	+	0.176853i
$902$		0			0
$903$	8130.11	−	7672.73i	0.299616	−	0.282760i
$904$		0			0
$905$			47529.8i			1.74579i
$906$		0			0
$907$	9786.28	−	9786.28i	0.358267	−	0.358267i	−0.504907	−	0.863174i	$-0.668473\pi$
$907$	9786.28	−	9786.28i	0.358267	−	0.358267i	0.863174	+	0.504907i	$0.168473\pi$
$908$		0			0
$909$	−6986.41	−	6986.41i	−0.254922	−	0.254922i
$910$		0			0
$911$		−	44848.6i		−	1.63107i	−0.578710	−	0.815533i	$-0.696444\pi$
$911$		−	44848.6i		−	1.63107i	0.578710	−	0.815533i	$-0.303556\pi$
$912$		0			0
$913$			14666.8i			0.531654i
$914$		0			0
$915$	−19899.4	+	19899.4i	−0.718965	+	0.718965i
$916$		0			0
$917$	2216.30	+	64.1450i	0.0798131	+	0.00230998i
$918$		0			0
$919$	−12886.8			−0.462564			−0.231282	−	0.972887i	$-0.574292\pi$
$919$	−12886.8			−0.462564			−0.231282	+	0.972887i	$0.574292\pi$
$920$		0			0
$921$		−	20362.1i		−	0.728507i
$922$		0			0
$923$	8040.90	+	8040.90i	0.286749	+	0.286749i
$924$		0			0
$925$	425.687	−	425.687i	0.0151314	−	0.0151314i
$926$		0			0
$927$	13897.2			0.492388
$928$		0			0
$929$			50751.1i			1.79235i	0.443704	+	0.896173i	$0.353664\pi$
$929$			50751.1i			1.79235i	−0.443704	+	0.896173i	$0.646336\pi$
$930$		0			0
$931$	3937.10	+	4421.33i	0.138596	+	0.155643i
$932$		0			0
$933$	−18657.0	−	18657.0i	−0.654665	−	0.654665i
$934$		0			0
$935$	−17134.2			−0.599305
$936$		0			0
$937$	−7509.51			−0.261820			−0.130910	−	0.991394i	$-0.541790\pi$
$937$	−7509.51			−0.261820			−0.130910	+	0.991394i	$0.541790\pi$
$938$		0			0
$939$	−2950.21	+	2950.21i	−0.102531	+	0.102531i
$940$		0			0
$941$	14978.5	+	14978.5i	0.518901	+	0.518901i	0.917239	−	0.398338i	$-0.130413\pi$
$941$	14978.5	+	14978.5i	0.518901	+	0.518901i	−0.398338	+	0.917239i	$0.630413\pi$
$942$		0			0
$943$	43948.7			1.51767
$944$		0			0
$945$	21426.4	+	22703.6i	0.737567	+	0.781534i
$946$		0			0
$947$	−24562.7	−	24562.7i	−0.842853	−	0.842853i	0.146376	−	0.989229i	$-0.453239\pi$
$947$	−24562.7	−	24562.7i	−0.842853	−	0.842853i	−0.989229	+	0.146376i	$0.953239\pi$
$948$		0			0
$949$	−20805.8	−	20805.8i	−0.711680	−	0.711680i
$950$		0			0
$951$		−	15581.0i		−	0.531281i
$952$		0			0
$953$		−	24293.4i		−	0.825751i	−0.910788	−	0.412875i	$-0.864524\pi$
$953$		−	24293.4i		−	0.825751i	0.910788	−	0.412875i	$-0.135476\pi$
$954$		0			0
$955$	17575.7	+	17575.7i	0.595536	+	0.595536i
$956$		0			0
$957$	−7425.07	−	7425.07i	−0.250803	−	0.250803i
$958$		0			0
$959$	−39500.7	+	37278.5i	−1.33008	+	1.25525i
$960$		0			0
$961$	−25306.7			−0.849476
$962$		0			0
$963$	1619.56	+	1619.56i	0.0541949	+	0.0541949i
$964$		0			0
$965$	−16740.1	+	16740.1i	−0.558428	+	0.558428i
$966$		0			0
$967$	−7370.14			−0.245096			−0.122548	−	0.992463i	$-0.539107\pi$
$967$	−7370.14			−0.245096			−0.122548	+	0.992463i	$0.539107\pi$
$968$		0			0
$969$	−5812.31			−0.192692
$970$		0			0
$971$	−28985.8	−	28985.8i	−0.957980	−	0.957980i	0.0411718	−	0.999152i	$-0.486891\pi$
$971$	−28985.8	−	28985.8i	−0.957980	−	0.957980i	−0.999152	+	0.0411718i	$0.986891\pi$
$972$		0			0
$973$	−132.924	+	4592.71i	−0.00437960	+	0.151321i
$974$		0			0
$975$		−	503.563i		−	0.0165404i
$976$		0			0
$977$	−39125.4			−1.28120			−0.640601	−	0.767874i	$-0.721315\pi$
$977$	−39125.4			−1.28120			−0.640601	+	0.767874i	$0.721315\pi$
$978$		0			0
$979$	13935.2	−	13935.2i	0.454923	−	0.454923i
$980$		0			0
$981$	13334.0	+	13334.0i	0.433967	+	0.433967i
$982$		0			0
$983$			16539.2i			0.536640i	0.963330	+	0.268320i	$0.0864685\pi$
$983$			16539.2i			0.536640i	−0.963330	+	0.268320i	$0.913531\pi$
$984$		0			0
$985$	−34155.2			−1.10485
$986$		0			0
$987$	35279.9	+	1021.08i	1.13776	+	0.0329296i
$988$		0			0
$989$	−20773.7	+	20773.7i	−0.667912	+	0.667912i
$990$		0			0
$991$		−	24591.8i		−	0.788277i	−0.919051	−	0.394139i	$-0.871043\pi$
$991$		−	24591.8i		−	0.788277i	0.919051	−	0.394139i	$-0.128957\pi$
$992$		0			0
$993$			9033.27i			0.288683i
$994$		0			0
$995$	24299.9	+	24299.9i	0.774229	+	0.774229i
$996$		0			0
$997$	−33317.6	+	33317.6i	−1.05835	+	1.05835i	−0.0601664	+	0.998188i	$0.519163\pi$
$997$	−33317.6	+	33317.6i	−1.05835	+	1.05835i	−0.998188	+	0.0601664i	$0.980837\pi$
$998$		0			0
$999$			42144.4i			1.33472i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.j.b.335.14		88
4.3	odd	2		112.4.j.b.27.8	yes	88
7.6	odd	2	inner	448.4.j.b.335.31		88
16.3	odd	4	inner	448.4.j.b.111.31		88
16.13	even	4		112.4.j.b.83.7	yes	88
28.27	even	2		112.4.j.b.27.7	✓	88
112.13	odd	4		112.4.j.b.83.8	yes	88
112.83	even	4	inner	448.4.j.b.111.14		88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.j.b.27.7	✓	88	28.27	even	2
112.4.j.b.27.8	yes	88	4.3	odd	2
112.4.j.b.83.7	yes	88	16.13	even	4
112.4.j.b.83.8	yes	88	112.13	odd	4
448.4.j.b.111.14		88	112.83	even	4	inner
448.4.j.b.111.31		88	16.3	odd	4	inner
448.4.j.b.335.14		88	1.1	even	1	trivial
448.4.j.b.335.31		88	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.83962 + 2.83962i) q^{3} +(-7.83669 + 7.83669i) q^{5} +(12.7115 + 13.4692i) q^{7} +10.8731i q^{9} +O(q^{10})\)	\(q+(-2.83962 + 2.83962i) q^{3} +(-7.83669 + 7.83669i) q^{5} +(12.7115 + 13.4692i) q^{7} +10.8731i q^{9} +(13.0368 - 13.0368i) q^{11} +(40.8123 + 40.8123i) q^{13} -44.5065i q^{15} +83.8552i q^{17} +(12.2047 - 12.2047i) q^{19} +(-74.3432 - 2.15167i) q^{21} +195.456 q^{23} +2.17256i q^{25} +(-107.545 - 107.545i) q^{27} +(-100.285 + 100.285i) q^{29} +66.9646 q^{31} +74.0394i q^{33} +(-205.170 - 5.93810i) q^{35} +(-195.938 - 195.938i) q^{37} -231.783 q^{39} +224.852 q^{41} +(-106.283 + 106.283i) q^{43} +(-85.2089 - 85.2089i) q^{45} -474.554 q^{47} +(-19.8379 + 342.426i) q^{49} +(-238.117 - 238.117i) q^{51} +(57.0387 + 57.0387i) q^{53} +204.331i q^{55} +69.3137i q^{57} +(483.525 + 483.525i) q^{59} +(-447.111 - 447.111i) q^{61} +(-146.451 + 138.213i) q^{63} -639.667 q^{65} +(-475.111 - 475.111i) q^{67} +(-555.022 + 555.022i) q^{69} +197.022 q^{71} -509.792 q^{73} +(-6.16925 - 6.16925i) q^{75} +(341.313 + 9.87841i) q^{77} -196.469i q^{79} +317.204 q^{81} +(-562.513 + 562.513i) q^{83} +(-657.147 - 657.147i) q^{85} -569.546i q^{87} +1068.91 q^{89} +(-30.9247 + 1068.49i) q^{91} +(-190.154 + 190.154i) q^{93} +191.289i q^{95} -295.588i q^{97} +(141.750 + 141.750i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 4 q^{7}+O(q^{10}) \) 88 * q + 4 * q^7	\( 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77} - 1952 q^{81} + 496 q^{85} + 1964 q^{91} - 112 q^{93} - 7592 q^{99}+O(q^{100}) \) 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Introduction

L-functions

Modular forms

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